I've posted on this topic before. We have 2 sons - one with a late June birthday and one with a late May brithday. They both attend the same montessori preschool, and the preschool includes both a pre-K and a K class. We plan to have them both go to K there for convenience purposes (daycare is included before and after class, having both kids at the same location for an extra year, smaller K class size - about 7-8 students with 2 teachers).
My oldest is 4 and in the pre-K program right now. So far, they've seen nothing to indicate that he is not on track socially or developmentally for public K. I think he'd do just fine if we went there, but I also know keeping him in the private K will give him one more year to mature socially. The difference between 5 and 6 year olds is pretty big. So he'll go from Private K to public 1st grade, unless we see some compelling academic reason to have him repeat K in public school.